Theorem: Suppose \(\A\) and \(\B\) are positive definite matrices satisfying \(\A \gg \B\). Then \(\A^{-1} \gl \B^{-1}\).
Proof: Consider the partitioned matrix
\[\M = \begin{bmatrix} \A & \I \\ \I & \B \end{bmatrix}.\]Noting that \(\M\) is positive definite if \(\A \gg \B\), we can conclude that \((\M^{-1})_{22}\) is also positive definite (as is its inverse). Using the Schur complement, we therefore have
\[\B^{-1} - \I\A^{-1}\I = \B^{-1}-\A^{-1} \gg \zero\]